Answered

Which of the following is true about the base $b$ of a logarithmic function?


A$b>0$ and $b=1$
B$b>0$ and $b

eq 1$

C$b<0$ and $b

eq 1$

D$b<0$ and $b=1$

Verified Answer :

  • The logarithmic function \(y = \log_b(x)\) is the inverse of the exponential function \(x = b^y\).
  • To define a valid inverse function, the exponential base \(b\) must be greater than zero (\(b > 0\)) to avoid non-real results and undefined values.
  • The base must also not equal one (\(b eq 1\)) to ensure the function is one-to-one and invertible.
  • The correct choice is \(\boxed{b > 0 \text{ and } b eq 1}\).

Explanation

  1. Defining the Logarithmic Function and Inverse Relationship

The logarithmic function \(y = \log_b(x)\) is defined as the inverse of the exponential function \(x = b^y\). For this inverse to be a well-defined, one-to-one function, the base \(b\) must satisfy specific conditions that allow the exponential function to be bijective (both injective and surjective) across its domain and range. We need to analyze these constraints to determine the valid set of values for \(b\).

  1. Analyzing the Condition for Positive Bases

First, we consider the condition \(b > 0\). If the base \(b\) were zero, the expression \(0^y\) would be undefined for \(y \leq 0\) and zero for \(y > 0\), failing to cover all positive real numbers. If \(b\) were negative, the value of \(b^y\) would not be a real number for many values of \(y\) (for example, \((-2)^{1/2}\) is not a real number). Therefore, to ensure that \(b^y\) produces a unique real value for every real exponent \(y\) and covers the domain of the logarithm (positive real numbers), the base must be positive, \(b > 0\).

  1. Analyzing the Condition Excluding Unity

Next, we consider the condition \(b eq 1\). If the base were \(b = 1\), the exponential function would be \(f(y) = 1^y = 1\), which is a constant function. A constant function is not one-to-one (injective), meaning multiple inputs yield the same output. Since a function must be one-to-one to have a well-defined inverse function, a logarithmic function cannot be defined with a base of \(1\). Thus, we must have \(b eq 1\). Combining these two requirements, the valid base for a logarithm must be strictly positive and not equal to one.

  1. Conclusion

Comparing our findings with the given choices:

  • \(b > 0\) and \(b = 1\) (Incorrect)
  • \(b > 0\) and \(b eq 1\) (Correct)
  • \(b < 0\) and \(b eq 1\) (Incorrect)
  • \(b < 0\) and \(b = 1\) (Incorrect)

The correct condition is that the base \(b\) must be positive and not equal to one, represented by \(\boxed{b > 0 \text{ and } b eq 1}\).

Examples

In data science and engineering, logarithms are often used to compress scales, such as in the Richter scale for earthquakes or the decibel scale for sound. Because logarithms are defined by their base, choosing a valid base (like 10 or the natural base \(e \approx 2.718\)) is essential for the function to accurately map input values to their corresponding logarithmic outputs. Understanding these restrictions ensures that mathematical models remain consistent and valid when analyzing exponential growth or decay.

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